I am worried about what will happen to various nations' economies relatively soon, long before the population actually halves, but I'm not worried that the fertility rate would continue on its trend as demographics change. Ignoring the potential second-order effects of economic collapse, wars over resources, etc., I think fertility rate would stabilize given that culture and genetics would by definition quickly become dominated by the people who do reproduce.

> culture and genetics would by definition quickly become dominated by the people who do reproduce

Yeah, but in how many generations? Four, five? That's a lot of collapse before we get to that point.